Task:
More formally, given a weighted directed graph
\(G=(V,E,W)\) , where \(V\)  and \(E\)  denote the set of nodes and
directed edges (arrows) connecting them. \(W\)  is the set of
edge weights corresponding to each directed edge
\(e = (u,v) \in E\) . The graph \(G\)  can be represented by a
weighted adjacency matrix
\(A \in {R}^{|V| \times |V|}\) , where the \(u,v\) -th
element in \(A\) , i.e.,
\(A_{u,v}\)  has a value which corresponds to the edge weight
in \(W\)  of the directed edge \((u,v)\) .
In general, node embedding methods try to minimize an objective
\(\min _Y L(f(A), g(Y))\) 
where \(Y \in {R}^{|V| \times d}\) , for \(d \ll |V|\)  is
a \(d\) -dimensional node embedding matrix;
\(f: {R}^{|V| \times |V|}  \over \longrightarrow  {R}^{|V|\times |V|}\)  is a transformation of the weighted adjacency
matrix; \(g: {R}^{|V| \times d}  \over \longrightarrow {R}^{|V| \times |V|} \)  is a pairwise edge function;
and \(L: {R}^{|V| \times |V|}  \over \longrightarrow {R}^{|V| \times |V|}\)  is a loss function.
Dataset:
For our empirical evaluation, we used the popular
Cora
dataset [1]}.
The Cora dataset is a research citation network
(directed) comprising of 2708 scientific publications
classified into one of seven categories. The citation
network consists of 5429 links. Each publication (vertex)
in the dataset is described by a \(0/1\) -valued word vector
indicating the absence/presence of the corresponding word
from the dictionary. The dictionary consists of 1433
unique words. Thus, each vertex has a corresponding binary
vector of length 1433.
Experimental Setup:
For our experiment, we only focused on a
subset of the categories, i.e., three categories, namely
Genetic_Algorithms (Label 0),
Probabilistic_Methods (Label 1), and
Reinforcement_Learning (Label 2).
We computed an edge weight for each edge as the
Jaccard distance between the vectors associated with
the start and terminal vertices of the edge.
Results:
In Figure REF , we notice that as we
increase the number of iterations \(H\)  in our method, we get
a larger dimensional feature vector which starts to achieve
better separation of topics / labels among the nodes in the
citation network. Therefore, nodes that represent a given
topic cluster together and also move away from topics that
are different. We see this separation improve as we vary \(H\) 
from 4 to 6. In order to visualize these \(H\) -dimensional
vectors representing the nodes in the Cora graph, we used
\(t\)-Distributed Stochastic Neighbor Embedding
(t-SNE) [2]}, which is a technique
for dimensionality reduction that is particularly well
suited for the visualization of high-dimensional datasets.

